On the error exponents of binary state discrimination with composite hypotheses

TL;DR

This paper investigates the conditions under which error exponents in binary state discrimination with composite hypotheses are equal to divergence measures, revealing cases where equality fails, especially in infinite-dimensional or quantum settings.

Contribution

The paper provides explicit examples where the inequality between error exponents and divergences fails and proves conditions for equality across classical and quantum scenarios.

Findings

Equality may fail in infinite-dimensional classical systems with countably infinite hypotheses.

In quantum systems, strict inequality is generic for non-commuting states.

Explicit examples demonstrate the divergence-error exponent relationship varies with system properties.

Abstract

The trade-off between the two types of errors in binary state discrimination may be quantified in the asymptotics by various error exponents. In the case of simple i.i.d. hypotheses, each of these exponents is equal to a divergence (pseudo-distance) of the two states. In the case of composite hypotheses, represented by sets of states $R,S$, one always has the inequality $\mathrm{e}(R\|S)\le \mathrm{E}(R\|S)$, where $\mathrm{e}$ is the exponent, $\mathrm{E}$ is the corresponding divergence, and the question is whether equality holds. The relation between the composite exponents and the worst pairwise exponents may be influenced by a number of factors: the type of exponents considered; whether the problem is classical or quantum; the cardinality and the geometric properties of the sets representing the hypotheses; and, on top of the above, possibly whether the underlying Hilbert space is finite- or infinite-dimensional. Our main contribution in this paper is clarifying this landscape considerably: We exhibit explicit examples for hitherto unstudied cases where the above inequality fails to hold with equality, while we also prove equality for various general classes of state discrimination problems. In particular, we show that equality may fail for any of the error exponents even in the classical case, if the system is allowed to be infinite-dimensional, and the alternative hypothesis contains countably infinitely many states. Moreover, we show that in the quantum case strict inequality is the generic behavior in the sense that, starting from any pair of non-commuting density operators of any dimension, and for any of the exponents, it is possible to construct an example with a simple null-hypothesis and an alternative hypothesis consisting of only two states, such that strict inequality holds for the given exponent.